Questions — OCR MEI FP1 (190 questions)

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OCR MEI FP1 2009 June Q8
8 Fig. 8 shows an Argand diagram. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fa71f270-53cb-44ba-b3a6-3953fa5c4232-3_421_586_1105_778} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Write down the equation of the locus represented by the perimeter of the circle in the Argand diagram.
  2. Write down the equation of the locus represented by the half-line \(\ell\) in the Argand diagram.
  3. Express the complex number represented by the point P in the form \(a + b \mathrm { j }\), giving the exact values of \(a\) and \(b\).
  4. Use inequalities to describe the set of points that fall within the shaded region (excluding its boundaries) in the Argand diagram.
OCR MEI FP1 2009 June Q9
9 You are given that \(\mathbf { M } = \left( \begin{array} { l l } 3 & 0
0 & 2 \end{array} \right) , \mathbf { N } = \left( \begin{array} { l l } 0 & 1
1 & 0 \end{array} \right)\) and \(\mathbf { Q } = \left( \begin{array} { r r } 0 & - 1
1 & 0 \end{array} \right)\).
  1. The matrix products \(\mathbf { Q } ( \mathbf { M N } )\) and \(( \mathbf { Q M } ) \mathbf { N }\) are identical. What property of matrix multiplication does this illustrate? Find QMN.
    \(\mathbf { M } , \mathbf { N }\) and \(\mathbf { Q }\) represent the transformations \(\mathrm { M } , \mathrm { N }\) and Q respectively.
  2. Describe the transformations \(\mathrm { M } , \mathrm { N }\) and Q . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{fa71f270-53cb-44ba-b3a6-3953fa5c4232-4_668_908_788_621} \captionsetup{labelformat=empty} \caption{Fig. 9}
    \end{figure}
  3. The points \(\mathrm { A } , \mathrm { B }\) and C in the triangle in Fig. 9 are mapped to the points \(\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime }\) and \(\mathrm { C } ^ { \prime }\) respectively by the composite transformation N followed by M followed by Q . Draw a diagram showing the image of the triangle after this composite transformation, labelling the image of each point clearly.
OCR MEI FP1 2010 June Q1
1 Find the values of \(A , B\) and \(C\) in the identity \(4 x ^ { 2 } - 16 x + C \equiv A ( x + B ) ^ { 2 } + 2\).
OCR MEI FP1 2010 June Q2
2 You are given that \(\mathbf { M } = \left( \begin{array} { r r } 2 & - 5
3 & 7 \end{array} \right)\).
\(\mathbf { M } \binom { x } { y } = \binom { 9 } { - 1 }\) represents two simultaneous equations.
  1. Write down these two equations.
  2. Find \(\mathbf { M } ^ { - 1 }\) and use it to solve the equations.
OCR MEI FP1 2010 June Q3
3 The cubic equation \(2 z ^ { 3 } - z ^ { 2 } + 4 z + k = 0\), where \(k\) is real, has a root \(z = 1 + 2 \mathrm { j }\).
Write down the other complex root. Hence find the real root and the value of \(k\).
OCR MEI FP1 2010 June Q4
4 The roots of the cubic equation \(x ^ { 3 } - 2 x ^ { 2 } - 8 x + 11 = 0\) are \(\alpha , \beta\) and \(\gamma\).
Find the cubic equation with roots \(\alpha + 1 , \beta + 1\) and \(\gamma + 1\).
OCR MEI FP1 2010 June Q5
5 Use the result \(\frac { 1 } { 5 r - 1 } - \frac { 1 } { 5 r + 4 } \equiv \frac { 5 } { ( 5 r - 1 ) ( 5 r + 4 ) }\) and the method of differences to find $$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 5 r - 1 ) ( 5 r + 4 ) }$$ simplifying your answer.
OCR MEI FP1 2010 June Q6
6 A sequence is defined by \(u _ { 1 } = 2\) and \(u _ { n + 1 } = \frac { u _ { n } } { 1 + u _ { n } }\).
  1. Calculate \(u _ { 3 }\).
  2. Prove by induction that \(u _ { n } = \frac { 2 } { 2 n - 1 }\). Section B (36 marks)
OCR MEI FP1 2010 June Q7
7 Fig. 7 shows an incomplete sketch of \(y = \frac { ( 2 x - 1 ) ( x + 3 ) } { ( x - 3 ) ( x - 2 ) }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e449d411-aaa9-4167-aa9c-c28d31446d52-3_786_1376_450_386} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Find the coordinates of the points where the curve cuts the axes.
  2. Write down the equations of the three asymptotes.
  3. Determine whether the curve approaches the horizontal asymptote from above or below for large positive values of \(x\), justifying your answer. Copy and complete the sketch.
  4. Solve the inequality \(\frac { ( 2 x - 1 ) ( x + 3 ) } { ( x - 3 ) ( x - 2 ) } < 2\).
OCR MEI FP1 2010 June Q8
8 Two complex numbers, \(\alpha\) and \(\beta\), are given by \(\alpha = \sqrt { 3 } + \mathrm { j }\) and \(\beta = 3 \mathrm { j }\).
  1. Find the modulus and argument of \(\alpha\) and \(\beta\).
  2. Find \(\alpha \beta\) and \(\frac { \beta } { \alpha }\), giving your answers in the form \(a + b \mathrm { j }\), showing your working.
  3. Plot \(\alpha , \beta , \alpha \beta\) and \(\frac { \beta } { \alpha }\) on a single Argand diagram.
OCR MEI FP1 2010 June Q9
9 The matrices \(\mathbf { P } = \left( \begin{array} { r r } 0 & 1
- 1 & 0 \end{array} \right)\) and \(\mathbf { Q } = \left( \begin{array} { l l } 2 & 0
0 & 1 \end{array} \right)\) represent transformations \(P\) and \(Q\) respectively.
  1. Describe fully the transformations P and Q . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e449d411-aaa9-4167-aa9c-c28d31446d52-4_625_849_470_648} \captionsetup{labelformat=empty} \caption{Fig. 9}
    \end{figure} Fig. 9 shows triangle T with vertices \(\mathrm { A } ( 2,0 ) , \mathrm { B } ( 1,2 )\) and \(\mathrm { C } ( 3,1 )\).
    Triangle T is transformed first by transformation P , then by transformation Q .
  2. Find the single matrix that represents this composite transformation.
  3. This composite transformation maps triangle T onto triangle \(\mathrm { T } ^ { \prime }\), with vertices \(\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime }\) and \(\mathrm { C } ^ { \prime }\). Calculate the coordinates of \(\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime }\) and \(\mathrm { C } ^ { \prime }\). T' is reflected in the line \(y = - x\) to give a new triangle, T".
  4. Find the matrix \(\mathbf { R }\) that represents reflection in the line \(y = - x\).
  5. A single transformation maps \(\mathrm { T } ^ { \prime \prime }\) onto the original triangle, T . Find the matrix representing this transformation.
OCR MEI FP1 2011 June Q1
1
  1. Write down the matrix for a rotation of \(90 ^ { \circ }\) anticlockwise about the origin.
  2. Write down the matrix for a reflection in the line \(y = x\).
  3. Find the matrix for the composite transformation of rotation of \(90 ^ { \circ }\) anticlockwise about the origin, followed by a reflection in the line \(y = x\).
  4. What single transformation is equivalent to this composite transformation?
OCR MEI FP1 2011 June Q2
2 You are given that \(z = 3 - 2 \mathrm { j }\) and \(w = - 4 + \mathrm { j }\).
  1. Express \(\frac { z + w } { w }\) in the form \(a + b \mathrm { j }\).
  2. Express \(w\) in modulus-argument form.
  3. Show \(w\) on an Argand diagram, indicating its modulus and argument.
OCR MEI FP1 2011 June Q3
3 The equation \(x ^ { 3 } + p x ^ { 2 } + q x + 3 = 0\) has roots \(\alpha , \beta\) and \(\gamma\), where $$\begin{gathered} \alpha + \beta + \gamma = 4
\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 6 \end{gathered}$$ Find \(p\) and \(q\).
OCR MEI FP1 2011 June Q4
4 Solve the inequality \(\frac { 5 x } { x ^ { 2 } + 4 } < x\).
OCR MEI FP1 2011 June Q5
5 Given that \(\frac { 3 } { ( 3 r - 1 ) ( 3 r + 2 ) } \equiv \frac { 1 } { 3 r - 1 } - \frac { 1 } { 3 r + 2 }\), find \(\sum _ { r = 1 } ^ { 20 } \frac { 1 } { ( 3 r - 1 ) ( 3 r + 2 ) }\), giving your answer as an exact fraction.
OCR MEI FP1 2011 June Q6
6 Prove by induction that \(1 + 8 + 27 + \ldots + n ^ { 3 } = \frac { 1 } { 4 } n ^ { 2 } ( n + 1 ) ^ { 2 }\). Section B (36 marks)
OCR MEI FP1 2011 June Q7
7 A curve has equation \(y = \frac { ( x + 9 ) ( 3 x - 8 ) } { x ^ { 2 } - 4 }\).
  1. Write down the coordinates of the points where the curve crosses the axes.
  2. Write down the equations of the three asymptotes.
  3. Determine whether the curve approaches the horizontal asymptote from above or below for
    (A) large positive values of \(x\),
    (B) large negative values of \(x\).
  4. Sketch the curve.
OCR MEI FP1 2011 June Q8
8 A polynomial \(\mathrm { P } ( z )\) has real coefficients. Two of the roots of \(\mathrm { P } ( z ) = 0\) are \(2 - \mathrm { j }\) and \(- 1 + 2 \mathrm { j }\).
  1. Explain why \(\mathrm { P } ( z )\) cannot be a cubic. You are given that \(\mathrm { P } ( z )\) is a quartic.
  2. Write down the other roots of \(\mathrm { P } ( z ) = 0\) and hence find \(\mathrm { P } ( z )\) in the form \(z ^ { 4 } + a z ^ { 3 } + b z ^ { 2 } + c z + d\).
  3. Show the roots of \(\mathrm { P } ( z ) = 0\) on an Argand diagram and give, in terms of \(z\), the equation of the circle they lie on.
OCR MEI FP1 2011 June Q9
9 The simultaneous equations $$\begin{aligned} & 2 x - y = 1
& 3 x + k y = b \end{aligned}$$ are represented by the matrix equation \(\mathbf { M } \binom { x } { y } = \binom { 1 } { b }\).
  1. Write down the matrix \(\mathbf { M }\).
  2. State the value of \(k\) for which \(\mathbf { M } ^ { - 1 }\) does not exist and find \(\mathbf { M } ^ { - 1 }\) in terms of \(k\) when \(\mathbf { M } ^ { - 1 }\) exists. Use \(\mathbf { M } ^ { - 1 }\) to solve the simultaneous equations when \(k = 5\) and \(b = 21\).
  3. What can you say about the solutions of the equations when \(k = - \frac { 3 } { 2 }\) ?
  4. The two equations can be interpreted as representing two lines in the \(x - y\) plane. Describe the relationship between these two lines
    (A) when \(k = 5\) and \(b = 21\),
    (B) when \(k = - \frac { 3 } { 2 }\) and \(b = 1\),
    (C) when \(k = - \frac { 3 } { 2 }\) and \(b = \frac { 3 } { 2 }\). RECOGNISING ACHIEVEMENT
OCR MEI FP1 2012 June Q1
1 You are given that the matrix \(\left( \begin{array} { r r } - 1 & 0
0 & 1 \end{array} \right)\) represents a transformation \(A\), and that the matrix \(\left( \begin{array} { r r } 0 & 1
- 1 & 0 \end{array} \right)\) represents a transformation B .
  1. Describe the transformations A and B .
  2. Find the matrix representing the composite transformation consisting of A followed by B .
  3. What single transformation is represented by this matrix?
OCR MEI FP1 2012 June Q2
2 You are given that \(z _ { 1 }\) and \(z _ { 2 }\) are complex numbers.
\(z _ { 1 } = 3 + 3 \sqrt { 3 } \mathrm { j }\), and \(z _ { 2 }\) has modulus 5 and argument \(\frac { \pi } { 3 }\).
  1. Find the modulus and argument of \(z _ { 1 }\), giving your answers exactly.
  2. Express \(z _ { 2 }\) in the form \(a + b \mathrm { j }\), where \(a\) and \(b\) are to be given exactly.
  3. Explain why, when plotted on an Argand diagram, \(z _ { 1 } , z _ { 2 }\) and the origin lie on a straight line.
OCR MEI FP1 2012 June Q3
3 The cubic equation \(3 x ^ { 3 } + 8 x ^ { 2 } + p x + q = 0\) has roots \(\alpha , \frac { \alpha } { 6 }\) and \(\alpha - 7\). Find the values of \(\alpha , p\) and \(q\).
OCR MEI FP1 2012 June Q4
4 Solve the inequality \(\frac { 3 } { x - 4 } > 1\).
OCR MEI FP1 2012 June Q5
5
  1. Show that \(\frac { 1 } { 2 r + 1 } - \frac { 1 } { 2 r + 3 } \equiv \frac { 2 } { ( 2 r + 1 ) ( 2 r + 3 ) }\).
  2. Use the method of differences to find \(\sum _ { r = 1 } ^ { 30 } \frac { 1 } { ( 2 r + 1 ) ( 2 r + 3 ) }\), expressing your answer as a fraction.